Find the equation of the normal to the curve `x = acostheta` and `y = b sintheta` at `theta`

Answer

To find the equation of the normal to the curve defined by the parametric equations \( x = a \cos \theta \) and \( y = b \sin \theta \) at a given angle \( \theta \), we will follow these steps:

### Step 1: Find the point on the curve at \( \theta \)

The coordinates of the point on the curve at \( \theta \) can be found using the parametric equations:

\[ x_1 = a \cos \theta \]
\[ y_1 = b \sin \theta \]

### Step 2: Find the derivatives \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \)

To find the slope of the tangent line, we need to calculate the derivatives of \( y \) and \( x \) with respect to \( \theta \):

\[ \frac{dy}{d\theta} = b \cos \theta \]
\[ \frac{dx}{d\theta} = -a \sin \theta \]

### Step 3: Calculate the slope of the tangent line

The slope of the tangent line \( m_t \) can be found using the formula:

\[ m_t = \frac{dy/d\theta}{dx/d\theta} = \frac{b \cos \theta}{-a \sin \theta} = -\frac{b}{a} \cot \theta \]

### Step 4: Find the slope of the normal line

The slope of the normal line \( m_n \) is the negative reciprocal of the slope of the tangent line:

\[ m_n = -\frac{1}{m_t} = \frac{a}{b} \tan \theta \]

### Step 5: Write the equation of the normal line

The general equation of a line in point-slope form is given by:

\[ y - y_1 = m_n (x - x_1) \]

Substituting \( y_1 = b \sin \theta \), \( x_1 = a \cos \theta \), and \( m_n = \frac{a}{b} \tan \theta \):

\[ y - b \sin \theta = \frac{a}{b} \tan \theta (x - a \cos \theta) \]

### Step 6: Simplify the equation

Rearranging the equation gives:

\[ y - b \sin \theta = \frac{a}{b} \tan \theta \cdot x - \frac{a^2}{b} \tan \theta \cdot \cos \theta \]

Multiplying through by \( b \) to eliminate the fraction:

\[ b(y - b \sin \theta) = a \tan \theta \cdot x - a^2 \tan \theta \cdot \cos \theta \]

This simplifies to:

\[ by - b^2 \sin \theta = ax \tan \theta - a^2 \sin \theta \]

Bringing all terms to one side gives:

\[ ax \tan \theta - by + b^2 \sin \theta - a^2 \sin \theta = 0 \]

### Final Equation of the Normal

Thus, the final equation of the normal line is:

\[ a x \tan \theta - b y = \sin \theta (a^2 - b^2) \]